On Weisfeiler-Leman Invariance: Subgraph Counts and Related Graph Properties

11/09/2018 ∙ by V. Arvind, et al. ∙ 0

We investigate graph properties and parameters that are invariant under Weisfeiler-Leman equivalence, focussing especially on 1-dimensional Weisfeiler-Leman equivalence and on subgraph counts that are preserved. Our main results are summarized below: -- For 1-dimensional Weisfeiler-Leman equivalence, we completely characterize graphs whose: (a) counts are determined by 1-dimensional Weisfeiler-Leman equivalence, and (b) whose presence is determined by 1-dimensional Weisfeiler-Leman equivalence. -- Extending an old result due to Beezer and Farrell for distance-regular graphs, we show that 2-dimensional Weisfeiler-Leman equivalence preserves the first five coefficients of the matching polynomial (and no more). We determine exactly which paths and cycles have their counts determined by 2-dimensional WL-equivalence. -- We also study a notion of "approximate invariance" of graph parameters under Weisfeiler-Leman equivalence. We show that the fractional matching number and fractional domination number are preserved by 1-dimensional Weisfeiler-Leman equivalence.

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